nLab quantization via the A-model

Idea

A method of quantization of mechanical systems represented by symplectic manifolds that produces the space of states of this system as the space of states of open strings in the A-model.

More in detail, the symplectic manifold serves as the target space for the 2-dimensional sigma-model TQFT called the A-model. If the symplectic manifold arises as the complexification of a phase space of some physical system, then one finds that the boundary path integral of the A-model on coisotropic branes computes the path integral and the geometric quantization of this physical system.

The setup is reminiscent of how the deformation quantization of the phase space is computed by the 2-dimensional Poisson sigma-model. See at holographic principle for more on the general pattern.

The goal is to get closer to a systematic theory of quantization.

(Gukov-Witten 08, p.4)

Applications

When the phase space $M$ is the coadjoint orbit for a real form $G_\mathbb{R}$, with the complexification being the complex coadjoint orbit $Y$ for $G_\mathbb{C}$. The particular cases when $G_\mathbb{C} = SL(2,\mathbb{C})$ are covered in section 3 of (Gukov-Witten 08). These provide applications to representation theory like in the orbit method.

Related concepts

• deformation quantization

• geometric quantization

References

The main original article is

• Sergei Gukov, Edward Witten, Branes and Quantization, Adv. Theor. Math. Phys. 13 (2009) 1–73, (arXiv:0809.0305, euclid)

Review includes

• Constantin Teleman, Branes and Representations,2016

Earlier, suggestions that the A-model is related to deformation quantization had been given in

• Paul Bressler, Yan Soibelman, Mirror symmetry and deformation quantization (arXiv:hep-th/0202128)

• Anton Kapustin, A-branes and Noncommutative Geometry (arXiv:hep-th/0502212)

which then have been extended and made more precise in

• Vasily Pestun, Topological strings in generalized complex space, Adv.Theor.Math.Phys.11:399-450,2007 (arXiv:hep-th/0603145)

and in

• Marco Gualtieri, Branes on Poisson varieties (arXiv:0710.2719)

in the context of generalized complex geometry.

Tha the space of states of open strings in the A-model can be interpreted as a quantization of the corresponding symplectic manifold had first been noticed in examples in

• Marco Aldi, Eric Zaslow, Coisotropic Branes, Noncommutativity, and the Mirror Correspondence, JHEP 0506 (2005) 019 (arXiv:hep-th/0501247)

The canonical coisotropic brane was used to elucidate these matters further in

• Anton Kapustin, Edward Witten, Electric-Magnetic Duality And The Geometric Langlands Program (arXiv:hep-th/0604151)

is the context of the Kapustin-Witten TQFT.

A relation to the path integral is discussed in

• Edward Witten, A New Look At The Path Integral Of Quantum Mechanics (arXiv:1009.6032)

Relation to the B-model via mirror symmetry is discussed in

• Sergei Gukov, Quantization via Mirror Symmetry (arXiv:1011.2218)

On its relation to geometric quantization:

• Davide Gaiotto, Edward Witten. Probing Quantization Via Branes (2021). (arXiv:2107.12251)

• Joshua Lackman, Geometric Quantization Without Polarizations [arXiv:2405.01513]